Download Unbounded absolute weak convergence in Banach lattices

UNBOUNDED ABSOLUTE WEAK CONVERGENCE IN BANACH
LATTICES
arXiv:1608.02151v5 [math.FA] 14 Feb 2017
OMID ZABETI
Abstract. The concepts of unbounded norm convergent nets and unbounded order
convergent ones in Banach lattices are considered and investigated in several recent
papers by Gao, Deng, and et al. In this note, taking idea from these notions, we consider the concept unbounded absolute weak convergence (uaw-convergence, for short)
in a Banach lattice. A net (xα ) in a Banach lattice E is said to be uaw-convergent
to x ∈ E if for each u ∈ E+ , the net (|xα − x| ∧ u) converges to zero weakly. We
investigate some properties of uaw-convergence and its relationship to other types of
unbounded convergent nets. In particular, we characterize order continuous Banach
lattices and reflexive Banach lattices in term of uaw-convergence.
1. introduction and preliminaries
Let us start with a few remarks on different kinds of unbounded convergent nets in
Banach lattices. Let E be a Banach lattice. A net (xα ) in E is said to be unbounded
order convergent ( uo-convergent, for short) to x ∈ E if for each positive u ∈ E,
the net (|xα − x| ∧ u) converges to zero in order. It is called unbounded norm
convergent ( un-convergent, in brief) if k|xα − x| ∧ uk → 0. These concepts have
been investigated in several papers by Gao, Deng, and et al ( see [DOT, Niu14, Niu16,
Niux14, KMT] for a detailed exposition on these notions).
In this paper, we consider a version of an unbounded convergent net in term of weak
convergence. Let E ba a Banach lattice. A net (xα ) ⊆ E is said to be unbounded
absolutely weakly convergent (uaw-convergent) to x ∈ E if for each u ∈ E+ ,
uaw
(|xα − x| ∧ u) → 0 weakly. We use the notation xα −−→ x for uaw-convergence. We see
that this kind of convergence can be viewed as an ” unbounded ” version of absolute
weak convergence in Banach lattices. We consider its relationship with other sorts of
unbounded convergences. We show that this type of convergence is topological and
we consider some topological aspects of it. In addition, we investigate some equivalent
statements for order continuity in a Banach lattice and whose dual space in term
Date: February 15, 2017.
2010 Mathematics Subject Classification. Primary: 46B42, 54A20. Secondary: 46B40.
Key words and phrases. Banach lattice, unbounded absolute weak convergence, unbounded absolute weak topology, order continuous Banach lattice, reflexive Banach lattice.
1
2
O. ZABETI
of uaw-convergence. Also, we characterize reflexive Banach lattices in term of uawconvergence, as well. For undefined notations and terminology used in this paper, we
refer the reader to [Abr02, Abr06, Nieb91]. All vector lattices in this note, are assumed
to be over the real scalar field ( R ).
2. main results
2.1. Basic results. First, we consider some elementary but useful facts about uawconvergence. We regularly use of two inequalities; (x + y) ∧ u ≤ x ∧ u + y ∧ u for
positive elements x, y, u ∈ E and the decomposition f = f + − f − for any functional
on E. The latter one ensures us when a convergence holds for positive functionals, it
is true for every element in E ∗ .
Lemma 1.
i. uaw-limit is unique.
uaw
uaw
uaw
ii. If xα −−→ x and yβ −−→ y, then axα + byβ −−→ ax + by, for any scalars a, b.
uaw
uaw
iii. If xα −−→ x, then yβ −−→ x, for every subnet (yβ ) of (xα ).
uaw
uaw
iv. If xα −−→ x, then |xα | −−→ |x|.
uaw
uaw
v. xα −−→ x iff (xα − x) −−→ 0.
uaw
uaw
Proof. (i). Suppose (xα ) is a net in Banach lattice E such that xα −−→ x and xα −−→
y. We must show that x = y. For each u ∈ E+ and for each f ∈ E+∗ , we have
f (|xα − x| ∧ u) → 0 and f (|xα − y| ∧ u) → 0. Then from the inequality
f (|x − y| ∧ u) 6 f (|xα − x| ∧ u) + f (|xα − y| ∧ u),
we observe that f (|x − y| ∧ u) = 0. Now using the Hahn-Banach theorem, we see that
k|x − y| ∧ uk = 0. Put u = |x − y| and we have the result.
The implications (ii), (iii), (iv), and (v) are straitforward.
The following proposition illustrates the relation between absolutely weakly convergent nets and uaw-convergent ones. In addition, this justifies the name unbounded
absolute weak convergence.
|σ|(E,E ∗ )
Proposition 2. Suppose E is a Banach lattice and (xα ) ⊆ E is such that xα −−−−−→
uaw
0, then xα −−→ 0. For order bounded nets, these two types of convergences agree.
The following example presents a sequence which is uaw-convergent but not absolutely weakly convergent.
Example 3. Let E = c0 and the sequence (an ) ⊆ c0 be defined via an = (0, . . . , n2 , 0, . . .),
in which n2 is appeared in the n − th place. Suppose ε > 0 is arbitrary and u = (un )
UNBOUNDED ABSOLUTE WEAK CONVERGENCE IN BANACH LATTICES
3
is a positive element of c0 . For sufficiently large n ∈ N, we have |un | < ε so that
uaw
kan ∧ uk < ε. In particular, this means that an −−→ 0. We claim that (an ) is not
absolutely weakly convergent to zero. Let ε = 12 and bi = i12 so that (bi ) ∈ ℓ1 . Then
1
(i)
Σ∞
i=1 an bi = 1 ≮ 2 .
Remark 4. Let E be an AM-space. Then, it can be easily seen that the lattice
operations are weakly sequentially continuous so that in this case, weak convergence
implies uaw-convergence. Also, suppose E is an atomic order continuous Banach
lattice. Then by [Nieb91, Proposition 2.5.23], the lattice operations are again weakly
sequentially continuous. Therefore for example, in ℓp for 1 ≤ p ≤ ∞, weak convergence
implies uaw-convergence. Note that when E is a non-atomic order continuous Banach
lattice, the above conclusion is not true, in general; for, by the example discussed after
the mentioned proposition, Lp ([0, 1]) does not have this property.
The following simple observation is useful in its own right and in the sense that it
shows off the power of uaw-convergence.
Lemma 5. Let E be a Banach lattice. Then every disjoint net in E is uaw-convergent
to zero.
w
Proof. Fix u ∈ E+ . The net (|xα |∧u) is order bounded and disjoint so that |xα |∧u −
→ 0.
uaw
This means that xα −−→ 0.
In the following lemma, we consider a relation between uaw-convergence and weak
convergence using of a quasi-interior point. It is analogous to [DOT, Lemma 2.11].
The proof is similar so that we leave out it.
uaw
Lemma 6. Let E be a Banach lattice with a quasi-interior point e. Then xα −−→ 0
w
iff |xα | ∧ e −
→ 0.
The following corollary is also similar to [DOT, Corollary 2.12].
Corollary 7. Let E be an order continuous Banach lattice with a weak unit e. Then
uaw
w
xα −−→ 0 iff |xα | ∧ e −
→ 0.
2.2. Uaw-convergence is topological. In this step, we show that uaw-convergence
in a Banach lattice is topological; note that un-convergence by [DOT, Section 7] is
topological whilst uo-convergence need not be topological, in general; see, e.g., [Ord66].
For each u ∈ E+ , for each ε > 0, and each f ∈ E+∗ , put
Vu,ε,f = {x ∈ E : f (|x| ∧ u) < ε}.
4
O. ZABETI
Let ℵ0 be the collection of all sets of this form. We show that ℵ0 is a base of neighuaw
borhoods of zero for some Hausdorff linear topology. It is obvious that xα −−→ 0 iff
every set of ℵ0 contains a tail of this net, hence the uaw-convergence is the convergence
induced by the mentioned topology.
First, note that every element in ℵ0 contains zero. Now, we show that for every two
elements of ℵ0 , their intersection is again in ℵ0 . Take Vu1 ,ε1 ,f1 and Vu2 ,ε2 ,f2 in ℵ0 . Put
ε = ε1 + ε2 , u = u1 ∨ u2 , and f = f1 + f2 . We claim that Vu,ε,f ⊆ Vu1 ,ε1,f1 ∩ Vu2 ,ε2 ,f2 .
Take any x ∈ Vu,ε,f . Then f (|x| ∧ u) < ε. Therefore,
f1 (|x| ∧ u1 ) ≤ f1 (|x| ∧ u) ≤ f (|x| ∧ u) < ε,
so that x ∈ Vu1 ,ε1,f1 . Similarly x ∈ Vu2 ,ε2 ,f2 .
It is not a hard job to see that Vu,ε,f + Vu,ε,f ⊆ Vu,2ε,f , so that for each U ∈ ℵ0 , there
is a V ∈ ℵ0 such that V + V ⊆ U. In addition, one may easily verify that for every
U ∈ ℵ0 and every scalar λ with |λ| ≤ 1, we have λU ⊆ U.
Now, we show that for each U ∈ ℵ0 and each y ∈ U, there exists V ∈ ℵ0 with
y + V ⊆ U. Suppose y ∈ Vu,ε,f . We have to find δ > 0, a non-zero v ∈ E+ , and a
non-zero positive functional g on E such that y + Vv,δ,g ⊆ Vu,ε,f .
Put v := u and g := f . Since y ∈ Vu,ε,f , we have f (|y|∧u) < ε. Put δ := ε−f (|y|∧u).
We claim that y + Vv,δ,g ⊆ Vu,ε,f . Pick x ∈ Vv,δ,g . We show that x + y ∈ Vu,ε,f and we
are done.
f (|x + y| ∧ u) ≤ f (|x| ∧ u) + f (|y| ∧ u) < f (|y| ∧ u) + δ = ε.
T
Finally, we show that this topology is Hausdorff. It is enough to show that ℵ0 =
{0}. On a contrary, suppose 0 6= x ∈ Vu,ε,f for all non-zero u ∈ E+ , for all ε > 0, and
for all f ∈ E+∗ . In particular, f (x) < ε. Since ε is arbitrary, it follows that f (x) = 0
for all functionals on E. Using the Hahn-Banach theorem, yields the desired result.
Now, we are looking for some relations between uaw-topology and absolute weak
topology. Note that uaw-topology and absolute weak topology are never equal, in
general. Suppose E is a Banach lattice. Choose a disjoint unbounded sequence (xn ) ⊆
uaw
E. By Lemma 5, xn −−→ 0; although (xn ) can not be weakly convergent. Nevertheless,
there are good news if we restrict our attention to bounded sets. First, we have two
lemmas which are similar to Lemma 2.1 and Lemma 2.2 of [KMT]. The proofs are
analogous so that we omit them.
Lemma 8. Vu,ε,f is either contained in [−u, u] or contains a non-trivial ideal.
Lemma 9. If Vu,ε,f is contained in [−u, u], then u is a strong unit.
UNBOUNDED ABSOLUTE WEAK CONVERGENCE IN BANACH LATTICES
5
Proposition 10. Suppose E is a Banach lattice. If a neighborhood of uaw-topology is
norm bounded, then E has a strong unit.
Proof. Suppose Vu,ε,f is contained in BE for some u ∈ E+ , ε > 0, and f ∈ E+∗ . By
Lemma 8, Vu,ε,f is contained in [−u, u]; hence by Lemma 9, u is a strong unit.
Theorem 11. Suppose Banach lattice E has a strong unit. Then uaw-topology agrees
with absolute weak topology on BE .
Proof. Suppose E has a strong unit. Thus E is lattice and norm isometric to C(K)
for some compact Hausdorff space K. W.O.L.G, we may assume that E = C(K).
uaw
w
Suppose xα −−→ 0. It follows that |xα | ∧ 1 −
→ 0. We conclude that |xα |(z) → 0 for
w
each z ∈ K. Since (xα ) is norm bounded, this shows that |xα | −
→ 0.
2.3. Uaw-convergence in relation with un- and uo-convergence. In this step,
we investigate some relations between uaw-convergence and other sorts of unbounded
convergences.
Remark 12. Let E be an order continuous Banach lattice. It is an easy job to see that
every uo-null net is uaw-null. Note that the hypothesis ” order continuity” is essential
and can not be dropped. Consider E = C([0, 1]). Define the sequence (fn ) ⊆ E via
uo
fn (0) = 1, fn ( n1 ) = fn (1) = 0, and linear between them. We claim that fn −→ 0 but
uo
fn 9 0 in the uaw-convergence. Fix g ∈ E+ . Indeed, fn ∧ g ≤ fn ↓ 0, so that fn −→ 0.
Put g ≡ 12 and consider the linear functional φ on E defined by φ(f ) = f (0). One can
easily verify that φ(fn ∧ g) = 21 .
Now, we are looking for situations under which uaw-convergence and un-convergence
agree.
Theorem 13. Suppose E is a Banach lattice. Then the following are equivalent.
i. E is order continuous.
uaw
un
ii. xα −−→ 0 ⇔ xα −→ 0 for every net (xα ) ⊆ E.
uaw
un
iii. xn −−→ 0 ⇔ xn −→ 0 for every sequence (xn ) ⊆ E.
Proof. (i) → (ii). Let E be an order continuous Banach lattice and (xα ) ⊆ E be a net
w
which is uaw-convergent to zero. For each positive u ∈ E, we have |xα | ∧ u −
→ 0. By
un
[Abr06, Theorem 4.17], k|xα | ∧ uk → 0; that is xα −→ 0.
(ii) → (iii). It is trivial.
6
O. ZABETI
(iii) → (i). Assume that (xn ) is a disjoint order bounded sequence in E. By Lemma
uaw
un
5, xn −−→ 0. By assumption, xn −→ 0. Since the sequence is order bounded, we
conclude that kxn k → 0.
Combining Theorem 5.3 in [DOT] and Theorem 13, we can characterize atomic order
continuous Banach lattices among all order continuous Banach lattices in the following.
Corollary 14. Suppose E is an order continuous Banach lattice. Then un-convergence,
uaw-convergence, and uo-convergence agree iff E is atomic.
2.4. Uaw-convergence and sublattices. In this part, we consider a version of [Niux14,
Lemma 3.4 and Lemma 4.5] in term of uaw-convergence ( see also Theorem 3.2 from
[Niu16] ). In fact, we show that uaw-convergence in an order continuous Banach lattice
is stable under passing to and from an ideal or a sublattice.
Proposition 15. Suppose E is an order continuous Banach lattice and I is an ideal
uaw
uaw
of E. For a net (xα ) ⊆ I, xα −−→ 0 in I iff xα −−→ 0 in E.
uaw
Proof. Suppose xα −−→ 0 in E. For each f0 ∈ I+∗ , by the Hahn-Banach theorem, there
exists f ∈ E ∗ such that f = f0 on I. Fix u ∈ I+ . Then f0 (|xα | ∧ u) = f (|xα | ∧ u) → 0.
uaw
For the converse, suppose I is an ideal in E and xα −−→ 0 in I. Note that for each
w
v ∈ I d , |xα | ∧ v = 0 so that for each u ∈ I + I d , |xα | ∧ u −
→ 0. By [Abr06, Theorem
d
1.36], I + I is order dense in E. Fix w ∈ E+ and f ∈ E+∗ . We have w ∧ u ↑ w in
which u ∈ (I + I d )+ with u ≤ w, so that
f (w ∧ u) ↑ f (w),
by order continuity of E. Given ε > 0. There is some u ∈ (I + I d )+ such that
f (w) − f (u ∧ w) < 2ε . Also, there exists some α0 with f (|xα | ∧ u ∧ w) < 2ε for each
α ≥ α0 . Thus by Birkhoff’s inequality, we have
f (|xα | ∧ w) − f (|xα | ∧ u ∧ w) ≤ f (w − u ∧ w).
This means that f (|xα | ∧ w) < ε and the proof is finished.
In the following proposition, we show that in an order continuous Banach lattice,
uaw-convergence is stable in sublattices. It is a variant of [ Niu16, Theorem 3.2].
Proposition 16. Suppose E is an order continuous Banach lattice and F is a sublattice
uaw
uaw
of E. Then for a net (xα ) ⊆ F , xα −−→ 0 in E iff xα −−→ 0 in F .
UNBOUNDED ABSOLUTE WEAK CONVERGENCE IN BANACH LATTICES
7
uaw
Proof. Assume that F is a sublattice of E and (xα ) ⊆ F . Suppose xα −−→ 0 in E,
f0 ∈ F+∗ , and u ∈ F+ . By the Hahn-Banach theorem, there exists f ∈ E ∗ with f = f0
on F , so that f0 (|xα | ∧ u) = f (|xα | ∧ u) → 0.
uaw
Now, suppose xα −−→ 0 in F . Assume that I is the ideal in E generated by F .
Fix f ∈ E+∗ and u ∈ I+ . There exists some v ∈ F+ with u ≤ v. Thus f (|xα | ∧ u) ≤
uaw
f (|xα | ∧ v) → 0. Therefore xα −−→ 0 in I. Applying Proposition 15, we have the
result.
Remark 17. Note that when E is order continuous, by Theorem 13, uaw-convergence
and un-convergence are the same. Thus, we can restate Proposition 15 and Proposition
16 in term of un-convergence, too. In this case, Proposition 16 can be obtained using
Theorem 13 and [KMT, Corollary 4.6] with a different method, independently. Also,
order continuity can not be dropped in the mentioned results. Consider E = ℓ∞ and
I = c0 . The standard basis (en ) is un-convergent to zero in I but it is not in E.
2.5. Some relations between uaw-convergence, order continuity, and reflexivity. A net (xα ) in a Banach lattice E is said to be uaw-Cauchy if the net (xα −xβ ),
uaw-converges to zero. Now, we consider some properties of uaw-Cauchy nets; in addition, we investigate some relations between un-Cauchy nets and uaw-Cauchy ones.
In prior to anything, we have two simple observations for uaw-convergence which can
be considered for any linear topology, too.
Lemma 18. Every uaw-convergent net in a Banach lattice E is uaw-Cauchy.
For the converse, the following is immediate.
Lemma 19. Suppose E is a Banach lattice and (xα ) is a uaw-Cauchy net which has
a uaw-convergent subnet. Then it is uaw-convergent.
It is not difficult to see that every un-Cauchy net is uaw-Cauchy. Note that by
Theorem 13, in an order continuous Banach lattice, un-Cauchy nets and uaw-Cauchy
ones agree. But these notions are not equivalent, in general. Consider the following
example.
1
<α<
Example 20. Put E = C([0, 1]). For each n ∈ N, choose reals α, β with n+1
1
β < n1 . Define the sequence (fn ) on E via fn (0) = fn (1) = fn ( n1 ) = fn ( n+1
) = 0,
fn (α) = fn (β) = n; constant between them and linear otherwise. It is easy to see that
uaw
(fn )′ s are disjoint, so that by Lemma 5, (fn − fm ) −−→ 0 when m, n are sufficiently
large. On the other hand, put g ≡ 1; one can easily verify that k|fm − fn | ∧ 1k ≥ 1.
In addition, this example presents a uaw-null sequence which is not un-convergent.
8
O. ZABETI
In the following theorem, we characterize order continuous Banach lattices in term
of uaw-convergence. It is a variant of [Niu14, Theorem 2.1]. A similar statement for
un-convergence has been obtained in [KMT, Theorem 8.1], independently.
Theorem 21. For a Banach lattice E, the following are equivalent.
i. E is order continuous.
ii. every order bounded uaw-Cauchy sequence in E is norm convergent.
iii. every order bounded uaw-convergent sequence in E is norm convergent.
uaw
w∗
iv. for every norm bounded sequence (x∗n ) ⊆ E ∗ , x∗n −−→ 0 implies that x∗n −→ 0.
|σ|(E ∗ ,E)
uaw
v. for every norm bounded sequence (x∗n ) ⊆ E ∗ , x∗n −−→ 0 implies that x∗n −−−−−→
0.
Proof. (i) → (ii). Suppose (xn ) is an order bounded uaw-Cauchy sequence in E. Using
Proposition 2 and [Abr06, Theorem 4.17], we conclude that it is norm Cauchy so that
norm convergent.
(ii) → (iii). It is trivial.
(iii) → (i). Suppose (xn ) is a disjoint order bounded sequence in E. By Lemma
uaw
5, xn −−→ 0. By assumption, it is norm convergent. On the other hand, since the
|σ|(E,E ∗ )
sequence is order bounded, we see that xn −−−−−→ 0. Now, an easy application of the
Hahn-Banach theorem results in kxn k → 0.
uaw
(i) → (iv). Suppose that (x∗n ) ⊆ E ∗ is a norm bounded sequence such that x∗n −−→ 0.
Without loss of generality, we may assume that kx∗n k ≤ 1. For every positive x ∈ E
and every ε > 0, by [Abr06, Theorem 4.18], there exists some 0 ≤ y ∗ ∈ E ∗ with
(|x∗n | ∧ y ∗ )(x) − |x∗n |(x) < ε,
w∗
for each n ∈ N. This means that x∗n −→ 0.
uaw
uaw
Since in a Banach lattice xn −−→ 0 if and only if |xn | −−→ 0, we conclude that
(iv) ↔ (v).
uaw
(iv) → (i). Suppose (x∗n ) is a norm bounded disjoint sequence in E ∗ . Then x∗n −−→ 0.
w∗
By assumption, x∗n −→ 0. Now, [Nieb91, Corollary 2.4.3] may apply to yield the desired
result.
Remark 22. It can be easily seen that we can restate Theorem 21, in terms of nets,
too. Also, note that order boundedness is essential in Theorem 21. Consider E = c0 .
uaw
Suppose (en ) is the standard basis for E. It is easy to see that en −−→ 0 but it is not
norm convergent. Keep in your mind that (en ) is not order bounded; nevertheless, E
is order continuous.
UNBOUNDED ABSOLUTE WEAK CONVERGENCE IN BANACH LATTICES
9
In this part, we characterize order continuity of the dual of a Banach lattice in term
of uaw-convergence. Surprisingly, the converse of Theorem 6.4 in [DOT] holds when
we replace un-convergence with uaw-convergence.
Theorem 23. For a Banach lattice E, the following are equivalent.
i. E ∗ is order continuous.
uaw
w
ii. For every norm bounded net (xα ) ⊆ E, xα −−→ 0 implies xα −
→ 0.
uaw
w
iii. For every norm bounded sequence (xn ) ⊆ E, xn −−→ 0 implies xn −
→ 0.
Proof. (i) → (ii). Suppose E ∗ is order continuous and (xα ) is a norm bounded uawnull net in E. W.L.O.G, we may assume that kxα k ≤ 1 for every index α. By [Abr06,
Theorem 4.19], for each ε > 0 and for each f ∈ E+∗ , there exists u ∈ E+ such that
f (|x| − |x| ∧ u) < ε whenever kxk ≤ 1. In particular, f (|xα | − |xα | ∧ u) < ε. Since
f (|xα | ∧ u) → 0 we conclude that f (|xα |) < ε provided that α is sufficiently large, so
that f (xα ) → 0. It follows that (xα ) is weakly convergent to zero.
(ii) → (iii). It is trivial.
uaw
(iii) → (i). Suppose (xn ) ⊆ E is a disjoint norm bounded sequence. Thus, xn −−→ 0.
w
By hypothesis, xn −
→ 0. This proves the claim.
Note that the result of Theorem 23 is not valid if we replace uaw-convergent sequences with uaw-Cauchy ones in the hypothesis. Consider E = c0 ; indeed E ∗ is order
continuous. Put un = Σni=1 ei , where (ei ) is the standard basis of E. It is not hard to see
that (un ) is norm bounded and uaw-Cauchy but not weakly convergent. Nevertheless,
There will be another results if we consider uaw-Cauchy nets.
Theorem 24. For a Banach lattice E, the following are equivalent.
i. E is reflexive.
ii. Every norm bounded uaw-Cauchy net in E is weakly convergent.
iii. Every norm bounded uaw-Cauchy sequence in E is weakly convergent.
Proof. (i) → (ii). Suppose (xα ) is a norm bounded uaw-Cauchy net in E. In view
of Theorem 13 and [KMT, Theorem 6.4], we conclude that (xα ) is uaw-convergent.
Therefore, using Theorem 23 yields the desired result.
(ii) → (iii). It is trivial.
(iii) → (i). First, we show that E is a KB-space. On a contrary, suppose not.
Therefore E contains a lattice copy of c0 . W.L.O.G, we may assume that c0 ⊆ E.
Note that the sequence un = Σni=1 ei , where ei is the standard basis of c0 , is norm
bounded and weakly Cauchy in c0 but not weakly convergent. In addition, by Remark
10
O. ZABETI
4, we conclude that it is absolutely weakly Cauchy. Fix f ∈ E+∗ , u ∈ E+ . Then
f (|um − un | ∧ u) ≤ f (|um − un |) = f0 (|um − un |) → 0, where f0 is the restriction
of f to c0 . This is a contradiction. Now, we claim that E ∗ is order continuous.
Otherwise, E contains a lattice copy of ℓ1 . The sequence (ei ) is a norm bounded uawCauchy sequence in ℓ1 which is not weakly convergent; using the Schur property. By
Proposition 16, we conclude this happens in E. This would complete the proof.
Combining this with Theorem 13, we obtain the following.
Corollary 25. For a Banach lattice E, the following are equivalent.
i. E is reflexive.
ii. E is order continuous and every norm bounded un-Cauchy net in E is weakly
convergent.
iii. E is order continuous and every norm bounded uaw-Cauchy sequence in E is
weakly convergent.
Observe that order continuity is necessary and can not be removed. Put E = ℓ∞ . By
[KMT, Theorem 2.3], un-topology and norm topology on E agree so that every norm
bounded un-Cauchy net in E is weakly convergent; nevertheless, E is not reflexive. In
addition, we have another result if we consider uo-Cauchy nets which is introduced in
[Niux14]. Recall that a net (xα ) in a Banach lattice E is uo-Cauchy if the net (xα − xβ )
is uo-convergent to zero in E.
Theorem 26. For a Banach lattice E, the following are equivalent.
i. E is reflexive.
ii. Every norm bounded uo-Cauchy net in E is weakly convergent.
iii. Every norm bounded uo-Cauchy sequence in E is weakly convergent.
Proof. (i) → (ii). Suppose (xα ) is a norm bounded uo-Cauchy net in E. By [Niux14,
Theorem 4.7], it is uo-convergent. Thus, by [Wick77, Theorem 5], it is weakly convergent.
(ii) → (iii). It is trivial.
(iii) → (i). Suppose not. Thus, E contains either a lattice copy of c0 or ℓ1 . Let
(ei ) be the standard basis of c0 . Indeed, the sequence (un ) defined via un = Σni=1 ei is a
uo-Cauchy sequence in c0 which is not weakly convergent. By [Niu16, Corollary 3.3],
this happens in E which is a contradiction. Also, the sequence (ei ) is uo-Cauchy in ℓ1
but not weakly convergent. Another use of [Niu16, Corollary 3.3], yields the desired
result.
UNBOUNDED ABSOLUTE WEAK CONVERGENCE IN BANACH LATTICES
11
Note that when E is an order continuous Banach lattice, every uo-convergent net is
uaw-convergent. So, we can prove the part (i) → (ii) of the preceding theorem using
Theorem 24, too.
Theorem 27. Suppose E is an order continuous Banach lattice. Then every norm
bounded uaw-Cauchy sequence in E ∗ is w ∗ -convergent.
Proof. Suppose (x∗n ) is a norm bounded uaw-Cauchy sequence in E ∗ . By Theorem
21 (iv), we conclude that (x∗n ) is a w ∗ -Cauchy sequence. Now, the Banach-Alaoglu
theorem may apply to yield the desired result.
The following theorem should be compared with [KMT, Theorem 8.4]. In particular,
it shows that less hypotheses are needed if we replace un-convergence with uaw-one;
in the sense that we need not order continuity of E ∗ to prove order continuity of E,
see also [Niu14, Theorem 3.4].
Theorem 28. For a Banach lattice E, the following are equivalent.
i. Every w ∗ -null net in E ∗ is uaw-null.
ii. E ∗ is atomic and both E and E ∗ are order continuous.
Proof. On a contrary, suppose E is not order continuous. By [Nieb91, Corollary 2.4.3],
there exists a disjoint norm bounded sequence (x∗n ) ⊆ E ∗ which is not w ∗ -null. So, we
can take a subsequence (x∗nk ), a vector x0 ∈ E, and a positive real ε with |x∗nk (x0 )| > ε
for each k. By the Banach-Alaoglu theorem, there are a subnet (gα ) of (x∗nk ) and
w∗
uaw
a g ∈ E ∗ such that gα −→ g so that gα −−→ g. On the other hand, by Lemma 5,
uaw
gα −−→ 0. This implies that g ≡ 0 which is in contradiction with |gα (x0 )| > ε. Also,
note that by the third part of [CW98, Theorem 3.1], E ∗ is atomic. Now, we show that
E ∗ is also order continuous. Suppose x∗n ↓ 0. We conclude that x∗n (x) → 0 for each
w∗
uaw
x ∈ E. This asserts that x∗n −→ 0. By the assumption, x∗n −−→ 0. Since the sequence
is order bounded, we conclude that (x∗n ) is weakly null. By the Dini’s theorem (see
[Abr06, Theorem 3.52]), observe that kx∗n k → 0. This would complete the proof.
Remark 29. First note that from the first part of the preceding theorem, we can
conclude that E is atomic. In addition, observe that the order continuity of E ∗ is
essential and can not be dropped. Put E = ℓ1 . It is easy to see that the sequence
(un ) ⊆ ℓ∞ defined via un = (0, . . . , 0, 1, . . .) with n zero terms is w ∗ -null. But, it is not
uaw-null. For, by Theorem 11, uaw-topology and absolute weak topology agree on the
unit ball of ℓ∞ . Thus if the sequence is uaw-null, it should be weakly null which is not
12
O. ZABETI
possible by the Dini’s theorem (see [Abr06, Theorem 3.52]). Note that in this case, E
is order continuous, both E and E ∗ are atomic but E ∗ is not order continuous.
Also, we can restate [KMT, Proposition 8.5] in term of uaw-convergence as follows.
Proposition 30. Suppose E is a Banach lattice whose dual space is atomic. Then the
following are equivalent.
|σ|(E ∗ ,E)
uaw
i. For every net (x∗α ) ⊆ E ∗ , if x∗α −−−−−→ 0, then x∗α −−→ 0.
|σ|(E ∗ ,E)
uaw
ii. For every sequence (x∗n ) ⊆ E ∗ , if x∗n −−−−−→ 0, then x∗n −−→ 0.
iii. E ∗ is order continuous.
Proof. (i) → (ii). It is trivial.
|σ|(E ∗ ,E)
(ii) → (iii). Suppose x∗n ↓ 0. It follows that x∗n −−−−−→ 0. By the assumption,
uaw
x∗n −−→ 0. since the sequence is order bounded, we conclude that x∗n → 0 weakly. Now,
the Dini’s theorem may apply to convince us that kx∗n k → 0.
(iii) → (i). It is a combination of Theorem 13 and [KMT, Proposition 8.5].
2.6. Some consequences of uaw-topology. Combining Theorem 13 with Proposition 5.3, Theorem 5.4, Proposition 6.2, and Theorem 6.4 in [ KMT], we obtain the
following.
Corollary 31. Suppose E is a non-atomic order continuous Banach lattice and W is
a zero neighborhood for uaw-topology. If W is convex, then W = E.
Corollary 32. Let E be an order continuous Banach lattice. Then uaw-topology is
locally convex iff E is atomic.
Corollary 33. Let E be an order continuous Banach lattice. Then E is uaw-complete
iff E is finite dimensional.
Corollary 34. Let E be an order continuous Banach lattice. Then BE is uaw-complete
iff E is a KB-space.
Considering Theorem 13 and [KMT, Theorem 3.2], we have the following.
Corollary 35. Suppose E is an order continuous Banach lattice. Then uaw-topology
is metrizable iff E has a quasi interior point.
Remark 36. Note that order continuity is essential in Corollary 35 and can not be
removed. Consider E = ℓ∞ . By Theorem 11, uaw-topology and absolute weak topology on BE are equal. On the other hand, BE is not weakly metrizable since E ∗ is not
separable. This implies that E can not be metrizable with respect to the uaw-topology.
UNBOUNDED ABSOLUTE WEAK CONVERGENCE IN BANACH LATTICES
13
We also have a variant of [KMT, Proposition 6.6] in term of uaw-convergence.
Proposition 37. Suppose E is a Banach lattice whose dual space is order continuous
and C is a closed convex norm bounded subset of E. Then C is uaw-closed.
uaw
Proof. Suppose xα −−→ x for a net (xα ) ⊆ C and a vector x ∈ E. By Theorem 23,
w
xα −
→ x. Since C is closed and convex, it is weakly closed. It follows that x ∈ C. Corollary 38. Let E be a reflexive Banach lattice and C be a closed convex norm
bounded subset of E. Then E is uaw-complete.
uaw
Proof. Suppose (xα ) is a uaw-Cauchy net in C. By Corollary 34, xα −−→ x for some
x ∈ E. By Proposition 37, we conclude that x ∈ C.
The point of the proof of the following result was kindly provided to us by Niushan
Gao. We state a proof for the sake of completeness.
Lemma 39. Suppose E is a Banach lattice. Then every order interval is uaw-compact
iff E is order continuous and atomic.
Proof. Suppose every order interval is uaw-compact. By Proposition 2 we see that
every order interval is weakly compact; asserting that E is order continuous. By [
Abr06, Theorem 4.17], we conclude that it is norm compact. This in turn means that
E is also atomic. The converse implication is a consequence of Corollary 14.
A variant of [KMT, Theorem 7.5] is the following.
Proposition 40. Suppose E is a Banach lattice. Then BE is uaw-compact iff E is
an atomic KB-space.
Proof. Let BE be uaw-compact. Since order intervals are norm bounded, by Lemma
39, E is order continuous and atomic. By combining [KMT, Theorem 7.5] and Theorem
13, we conclude that E is a KB-space, too. The converse is essentially [KMT, Theorem
7.5] accompanying Theorem 13.
Remark 41. Consider this point that if E is non-atomic, the conclusion of Proposition
40 is not true, in general. Put E = L1 [0, 1]. It is an easy job to see that there is no
subsequence of the Rademacher functions (rn ) which is uaw-convergent.
14
O. ZABETI
3. acknowledgement
This note would not have existed without inspiring and invaluable suggestions and
comments of V. G. Troitsky, my friend and my colleague. I would like to have a deep
gratitude toward him. Special thanks is also due to Niushan Gao and Foivos Xanthos
for valuable comments.
References
[Abr02] Y. Abramovich and C.D. Aliprantis, An invitation to Operator theory, Vol. 50. Providence,
RI: American Mathematical Society, 2002.
[Abr06] C.D. Aliprantis and O. Burkinshaw, Positive operators, Springer, 2006.
[CW98] Z.L. Chen and A. W. Wickstead, Some applications of Rademacher sequences in Banach
lattices, Positivity., 2(1998), pp 171–191.
[DOT] Y. Deng, M O’Brien, and V. G. Troitsky, Unbounded norm convergence in Banach lattices,
Positivity., to appear. arXiv:1605.03538.
[Niu14] N. Gao, Unbounded order convergence in dual spaces, J. Math. Anal. Appl., 419(2014), pp
347-354.
[Niu16] N. Gao, V. G. Troitsky, and F. Xanthos, Uo-convergence and its applications to Cesro means
in Banach lattices, Israel J. Math., to appear. arXiv:1509.07914.
[Niux14] N. Gao and F. Xanthos, Unbounded order convergence and application to martingales without
probability, J. Math. Anal. Appl., 415 (2014), pp 931–947.
[KMT] M. Kandić, M.A.A. Marabeh, and V. G. Troitsky, Unbounded norm topology in Banach lattices, preprint. arXiv:1608.05489.
[Nieb91] P. Nieberg, Banach lattices, Springer-Verlag, Berlin, 1991.
[Ord66] E. T. Ordman, Convergence almost everywhere is not topological, American Math. Monthly.,
73(2), 1966, 182–183.
[Wick77] A. W. Wickstead, Weak and unbounded order convergence in Banach lattices, J. Austral.
Math. Soc., Ser.A 24(1977)312-319.
(O. Zabeti) Department of Mathematics, Faculty of Mathematics, University of Sistan and Baluchestan, Zahedan, P.O. Box 98135-674. Iran
E-mail address: [email protected]